Interrigator chaplain would be up there with Sanguinary Guard…
First off, the likelyhood of one either of them going to the Deathwatch while holding that rank (or being promoted to those ranks in the middle of a watch) is so rediculously miniscule its practically nil.
On the absolutely crazy-bat-poop chance it would happen (meaning all the stars in the galaxy align and your character makes a daemonic pact with tzeentch) it would be based off of an existing specialty (which is what its called in DW and sounds better than "class" anyways) with marks of distinction to set them apart. Any special wargear (none really for an Interagator chaplain) would have to be acquired through normal means for that speciality (Sanguinary Guard would have to get artificer armour, winged jump pack, angelus bolter, and blade encarmine, interagator chaplain… well he would need chaplain specialty and thus would get all the chaplain stuff).
I need to stress again, there is no reason either of these specialties would ever be secunded to the deathwatch. They have their specific chapter roles they need to fulfil for their chapter. If it ever happened it would be (rare as above) a very short secundment and for a very specific reason for the chapter itself (selfish motives vs. just joining up to do your bit).